- Level: AS and A Level
- Subject: Science
- Word count: 1397
Gravitation - Kepler and Newton revision notes and calculations.
Extracts from this document...
Introduction
Gravitation
Kepler (1571-1630) has studied for many years the records of observation on planets and summarised three laws.
Kepler’s Law
- Each planet moves in an ellipse which has the sun at one focus
- The line joining the sun and the moving planet sweeps out equal area in equal time
- the square of the time of revolution of any planet (i.e. T) about the sun is proportional to the cube of the planets’ mean distance from the sun
I.e. is a constant
Interpretation form Kepler’s laws
- Kepler’s second law :
The area swept out in a very short time interval (Δt), neglecting the
small triangular region is A
The rate of area swept =
Hence it is a constant.
Compare this equation with the angular momentum
= Constant
This law is in fact an evidence of conservation of angular momentum.
- Kepler’s third law
About 1666, Newton investigated the motion of the moon, and thought that it was the force of gravity to pull the moon and keep it in its orbit
NB: Time between full moon is 29.5 days but this due to the earth also moving round the sun, the moon is therefore to travel a bit longer
At the earth surface g=9.81m . This is due to the fact that we are nearer the earth centre than the moon (about 1:60)
Middle
= g - (< g )
- At latitude θ
The body describes a circle of radius r centre
Note:
- Note that mCosθ is the resultant force of mg and m.
The direction of (or T) is not exactly towards the centre of the earth
except at the poles and theequator
2) Practical Values
=====================================
Latitude / m
9.78
9.79
9.82
9.83
=====================================
Variation of g with height
(i) r
&
➔ ➔
Let r = + h
Then
If h << then
(ii) r < (We assume that the earth is a sphere of uniform density)
Consider a small mass placed at A, the spherical shell does not produce
and gravitational field on the point inside it.
&
Since we assume that the earth has uniform density
∝ r
Graphical representation
Practical data
==================================================
Altitude/ h(m) /(m)
0 9.81
1000 9.80
5 8.53
1 7.41(Parking Orbit)
3.8 0.0027 (Radius of moon orbit)
==================================================
Satellite orbit
The satellite can state in a fixed orbit with a specified radius, r, and
corresponding velocity v by F = ma
--------------- (1)
since
---------------- (2)
(2) → (1):
Note
For a satellite closed to the earth, stay at a height of about 100 –200km
Then r
Parking Orbit
The earth is rotating, at this orbit; the satellite can stay constantly
Overhead of the earth
Angular velocity of the earth
F = ma
=
h =
Note:
Conclusion
radius R. Determine the gravitational potential at a point distance r
from centre, where r < R.
Solution:
When r < R
➔ ➔
When r = R
➔
E.g.7 The masses of the earth and the moon are respectively
and and they are separated by a
distance
- Ignore the motions the earth and the moon , sketch the
gravitational potential V along the line joining them.
- Find the point where the net gravitational field strength (g) is
zero.
Solution:
- Choose the earth to be the origin
V = Potential of earth + Potential of moon = +
Near the earth, V is dominated by the field of the earth and near the
moon, V is dominated by the field of the moon.
E.g.8 Some satellite, with the necessary electronic equipment inside, rises
vertically from the equator when it is fired. At a particular height
the satellite is given a horizontal momentum by firing rocket on its
surface and the satellite then turns into the required orbit.
A satellite is to be put into orbit 500km above the earth’s surface.
If it’s vertical velocity after launching is 2000 at this height.
Calculate the impulse required to put the satellite directly into
Orbit, if its mass is 50kg. (g = 10 Radius of earth =6400km)
solution:
Suppose u is the velocity required for the orbit, radius R. Then
Force on the satellite =
u = 7700
Let the impulse P required has the relation shown such that
END
This student written piece of work is one of many that can be found in our AS and A Level Mechanics & Radioactivity section.
Found what you're looking for?
- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month